Exponentially quasi-mixing limits for killed symmetric Lévy processes
Pub. online: 16 September 2025
Type: Research Article
Open Access
Open Access
1
The research of the first author was supported by Postgraduate Scientific Research Innovation Project of Xiangtan University (No. XDCX2023Y145).
Received
13 March 2025
13 March 2025
Revised
19 June 2025
19 June 2025
Accepted
29 August 2025
29 August 2025
Published
16 September 2025
16 September 2025
Abstract
Quasi-mixing limits of the killed symmetric Lévy process are studied. It is proved that (intrinsic) ultracontractivity of the underlying process implies the existence of its (uniformly) exponentially quasi-mixing limits. As a by-product, this implication ensures that the process has (uniformly) exponential quasi-ergodicity and (uniformly) exponentially fractional quasi-ergodicity on ${L^{p}}$ ($p\ge 1$). It is noteworthy that precise rates of convergence and precise limiting equalities are provided, which are determined by spectral gaps and eigenfunction ratios of the underlying process. Finally, three examples are provided to demonstrate the theoretical results.
References
Declarations
Conflict of interest. The authors declare no conflict of interest.
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Consent for Publication. The authors declare consent for publication.
Human and Animal Ethics. Not applicable.
Berrhazi, B.E., Bouggar, D., Mrhardy, F.N.: Reflected backward doubly stochastic differential equations driven by Teugels martingales associated to a Lévy process with discontinuous barrier. Markov Process. Relat. 29, 329–346 (2023). MR4683711
Benaïm, M., Champagnat, N., Oçafrain, W., Villemonais, D.: Transcritical bifurcation for the conditional distribution of a diffusion process. J. Theor. Probab. 36, 1555–1571 (2022). MR4621075. https://doi.org/10.1007/s10959-022-01216-7
Champagnat, N., Villemonais, D.: General criteria for the study of quasi-stationarity. Electron. J. Probab. 28, 1–84 (2023). MR4546021. https://doi.org/10.1214/22-ejp880
Champagnat, N., Villemonais, D.: Lyapunov criteria for uniform convergence of conditional distributions of absorbed Markov processes. Stoch. Process. Appl. 135, 51–74 (2021). MR4222402. https://doi.org/10.1016/j.spa.2020.12.005
Chen, J.W., Li, H.T., Jian, S.Q.: Some limit theorems for absorbing Markov processes. J. Phys. A, Math. Theor. 45 (2012). MR2960634. https://doi.org/10.1088/1751-8113/45/34/345003
Chen, J.W., Jian, S.Q.: A remark on quasi-ergodicity of ultracontractive Markov processes. Stat. Probab. Lett. 87, 184–190 (2014). MR3168953. https://doi.org/10.1016/j.spl.2014.01.006
Chen, Z.Q., Zhang, X.C., Zhao, G.H.: Supercritical SDEs driven by multiplicative stable-like Lévy processes. Trans. Am. Math. Soc. 374, 7621–7655 (2021). MR4328678. https://doi.org/10.1090/tran/8343
Chen, Z.Q., Song, R.M.: Intrinsic Ultracontractivity and Conditional Gauge for Symmetric Stable Processes. J. Funct. Anal. 150, 204–239 (1997). MR1473631. https://doi.org/10.1006/jfan.1997.3104
Davies, E.B.: Heat Kernels and Spectral Theory. Cambridge University Press, (1989). MR0990239. https://doi.org/10.1017/CBO9780511566158
Davies, E.B., Simon, B.: Ultracontractive semigroups and some problems in analysis. In: Aspects of Mathematics and Its Applications. North-Holland, Amsterdam (1984). MR0849557. https://doi.org/10.1016/S0924-6509(09)70260-0
Davies, E.B.: Linear operators and their spectra. Cambridge University Press, (2007). MR2359869. https://doi.org/10.1017/CBO9780511618864
Davies, E.B., Simon, B.: Ultracontractivity and the heat kernel for Schrödinger operators and Dirichlet Laplacians. J. Funct. Anal. 59, 335–395 (1984). MR0766493. https://doi.org/10.1016/0022-1236(84)90076-4
Guillin, A., Nectoux, B., Wu, L.M.: Quasi-stationary distribution for Hamiltonian dynamics with singular potentials. Probab. Theory Relat. Fields 185, 921–959 (2023). MR4556285. https://doi.org/10.1007/s00440-022-01154-9
Grzywny, T.: Intrinsic ultracontractivity for Lévy processes. Probab. Math. Stat-Pol. 28, 91–106 (2008). MR2445505
He, G.M., Zhang, H.J.: Exponential convergence to a quasi-stationary distribution for birth-death processes with an entrance boundary at infinity. J. Appl. Probab. 59, 983–991 (2022). MR4507677. https://doi.org/10.1017/jpr.2021.100
Journel, L., Monmarché, P.: Convergence of a particle approximation for the quasi-stationary distribution of a diffusion process: Uniform estimates in a compact soft case. ESAIM, Probab. Stat. 26, 1–25 (2022). MR4363453. https://doi.org/10.1051/ps/2021017
Kaleta, K., Lőrinczi, J.: Pointwise eigenfunction estimates and intrinsic ultracontractivity-type properties of Feynman-Kac semigroups for a class of Lévy processes. Ann. Probab. 43, 1350–1398 (2015). MR3342665. https://doi.org/10.1214/13-AOP897
Kim, P., Song, R.M.: Intrinsic ultracontractivity for non-symmetric Lévy processes. Forum Math. 21, 43–66 (2009). MR2494884. https://doi.org/10.1515/FORUM.2009.056
Knobloch, R., Partzsch, L.: Uniform Conditional Ergodicity and Intrinsic Ultracontractivity. Potential Anal. 33, 107–136 (2010). MR2658978. https://doi.org/10.1007/s11118-009-9161-5
Li, H.S., Zhang, H.J., Liao, S.X.: On quasi-stationaries for symmetric Markov processes. J. Math. Anal. Appl. 528 (2023). MR4604336. https://doi.org/10.1016/j.jmaa.2023.127498
Li, Y.X., Cohen, D., Kottos, T.: Enforcing Lévy relaxation for multi-mode fibers with correlated disorder. New J. Phys. 24 (2022). MR4455508. https://doi.org/10.1088/1367-2630/ac6318
Li, Z., Feng, T.Q., Xu, L.P.: Non-confluence of fractional stochastic differential equations driven by Lévy process. Fract. Calc. Appl. Anal. 27, 1414–1427 (2024). MR4746851. https://doi.org/10.1007/s13540-024-00278-0
Lee, J.: Analytic form of the quasi-stationary distribution of a simple birth-death process. J. Korean Phys. Soc. 77, 457–462 (2020). https://doi.org/10.3938/jkps.77.457
Leliévre, T., Ramil, M., Reygner, J.: Quasi-stationary distribution for the Langevin process in cylindrical domains, Part I: Existence, uniqueness and long-time convergence. Stoch. Process. Appl. 144, 173–201 (2022). MR4347490. https://doi.org/10.1016/j.spa.2021.11.005
Méléard, S., Villemonais, D.: Quasi-stationary distributions and population processes. Probab. Surv. 9, 340–410 (2012). MR2994898. https://doi.org/10.1214/11-PS191
Saloff-Coste, L.: Aspects of Sobolev-Type Inequalities. Cambridge University Press, (2001). MR1872526
Zhang, X.L., Zhang, X.C.: Ergodicity of supercritical SDEs driven by α-stable processes and heavy-tailed sampling. Bernoulli 29, 1933–1958 (2023). MR4580902. https://doi.org/10.3150/22-bej1526
Zhang, H.J., Li, H.S., Liao, S.X.: Intrinsic ultracontractivity and uniform convergenceto the Q-process for symmetric Markov processes. Electron. Commun. Probab. 28, 1–12 (2023). MR4684057. https://doi.org/10.1214/23-ecp550
Zhang, J.F., Li, S.M., Song, R.M.: Quasi-stationarity and quasi-ergodicity of general Markov processes. Sci. China, Math. 57, 2013–2024 (2014). MR3247530. https://doi.org/10.1007/s11425-014-4835-x
